Comprehensive Model of Single Particle Pulverized Coal Combustion


Feb 22, 2017 - The data were for two subbituminous coals (Black Thunder and North Antelope) and two high volatile bituminous coals (Utah Skyline and ...
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Comprehensive Model of Single Particle Pulverized Coal Combustion Extended to Oxy-Coal Conditions Troy Holland and Thomas H. Fletcher* Chemical Engineering Department, Brigham Young University, Provo, Utah 84602, United States ABSTRACT: Oxy-fired coal combustion is a promising potential carbon capture technology. Predictive CFD simulations are valuable tools in evaluating and deploying oxy-fuel and other carbon capture technologies either as retrofit technologies or for new construction. However, accurate predictive simulations require physically realistic submodels with low computational requirements. In particular, comprehensive char oxidation and gasification models have been developed that describe multiple reaction and diffusion processes. This work extends a comprehensive char conversion code (CCK), which treats surface oxidation and gasification reactions as well as processes such as film diffusion, pore diffusion, ash encapsulation, and annealing. In this work several submodels in the CCK code were updated with more realistic physics or otherwise extended to function in oxy-coal conditions. Improved submodels include the annealing model, the swelling model, the mode of burning parameter, and the kinetic model, as well as the addition of the chemical percolation devolatilization (CPD) model. Results of the char combustion model are compared to oxy-coal data, and further compared to parallel data sets near conventional conditions. A potential method to apply the detailed code in CFD work is given.

1. INTRODUCTION Coal-fired power plants have provided a substantial percentage of global electricity for decades, and current outlooks indicate that they will continue to do so for the foreseeable future. The high proportion of electrical power generation is matched by a correspondingly high proportion of CO2 emissions. In order to meet regulatory targets for reduced emissions, carbon capture and sequestration techniques must be employed, and oxy-coal combustion is a promising potential solution. Oxy-coal combustion has been reviewed thoroughly elsewhere,1,2 but in essence it consists of injecting high purity O2 with the pulverized coal rather than the conventional air-fired method. To reduce the boiler temperatures to manageable levels, the flue gas is typically recycled, producing a combustion environment with high concentrations of CO2, O2, and (potentially) H2O. The flue gas then contains very high concentrations of CO2, and the CO2 is thus relatively easy to capture. While an oxy-coal system simplifies carbon capture, it also radically changes the environment the coal particles experience. The new environment changes the O2 diffusion rate, may cool the char particle via endothermic gasification, and may alter the overall char consumption rate due to gasification reactions.3 These effects and others, such as reduced flame temperature, delayed ignition, decreased acid gases, and increased gas emissivity, can largely be ascribed to differences between CO2 and N2 (the respective diluents in oxy-coal and air-fired pulverized coal systems).1 The change in diluent gas induces several interrelated effects that alter the burnout time and radiative behavior of the system, so accurate CFD predictions of oxy-coal combustion require models that describe these phenomena. This work supports computational fluid dynamics (CFD) modeling of oxy-coal boilers either for the retrofit of existing boilers or the construction of new oxy-coal-fired power plants by providing a detailed code that can predict the temperature and burnout profiles of coal particles in a hot, © XXXX American Chemical Society

oxidative environment. The detailed model could also be used to train low computational cost, reduced-order models to accurately describe a specific scenario.

2. EXPERIMENTAL SECTION To conduct a relevant comparison, the model was executed at conditions related to real-world application. Here, the most applicable conditions are the oxy-coal combustion environment, so experimental data from the literature were chosen for comparison at useful conditions. The experimental data also allowed the kinetic parameters to be calibrated. The model was then compared both to the calibration data and to similar data not used in the calibration. The experimental data referenced here were collected by Shaddix and Molina4 and Geier et al.5 The reactor consists of a burner-stabilized flat flame, a quartz chimney for gas and particles to flow through, and a coal particle inlet in the center of the burner. The particle temperatures were measured with a 2-color pyrometry system, and the diameters were measured by imaging of the particle emission. No burnout data from probe measurements were available from this data set. The coal particle flow rate was sufficiently low that particles did not affect each other or the bulk gas composition. The data were for two subbituminous coals (Black Thunder and North Antelope) and two high volatile bituminous coals (Utah Skyline and Pittsburgh seam (Bailey)) which were subjected to conditions of 14 or 16% H2O, 12, 24, or 36% O2, and the balance CO2, at gas temperatures ranging from approximately 1400−1700 K. The proximate and ultimate analyses of the coals and a summary of experimental conditions are given in Table 1 and Table 2. The char particles were in the reactor for up to approximately 0.2 s (post devolatilization), and on the order of 1,000 particle data triplets of temperature, location, and diameter were collected for each condition. These data were used in a related sensitivity analysis of the carbon conversion kinetics (CCK) model6 to determine which model parameters were most sensitive at oxy-fuel conditions, and to target model updates and refinements. These Received: December 20, 2016 Revised: February 14, 2017 Published: February 22, 2017 A

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Energy & Fuels Table 1. Proximate and Ultimate Analysis of Coal Particles between 76 and 105 μm

a

Coal

Moisture (% AR)

Ash (% AR)

Volatiles (% AR)

C (% daf)

H (% daf)

O (% daf)a

N (% daf)

S (% daf)

Black Thunder Utah Skyline Pittsburgh North Antelope

9.34 1.69 0.47 10.83

4.84 10.2 6.95 5.54

42.34 40.79 35.89 39.64

68.96 79.4 81.26 72.12

5.00 6.09 5.55 5.45

25.41 12.25 10.17 21.08

0.97 1.67 1.54 1.00

0.45 0.59 2.16 0.35

By difference.

same parameter subset in all cases, almost entirely independent of combustion conditions or coal type.7 Though the sensitivity analysis used half of the same data cited in the Experimental Section, the sensitivity analysis only indicated which submodels were most influential. In the present work, those models are updated, based solely on recent research and literature observations, and completely independent of the char burnout data. Thus, this model was constructed to more precisely capture the physics of char burnout and not merely to fit the selected data set. The char burnout data influenced the model only in the final stage of kinetic parameter calibration. A complete list of parameters is given by Holland.9 Figure 1 is a

Table 2. Summary of Experiments for Char Particles between 53 and 125 μm Coal Black Thunder Pittsburgh

Utah Skyline North Antelope

O2 Mol %

CO2 Mol %

H2O Mol %

Peak Particle Temp (K)

Peak Gas Temp (K)

12 24 36 12 24 36 12 24 36 12 24 36

74 62 50 74 62 50 72 60 48 72 60 48

14 14 14 14 14 14 16 16 16 16 16 16

1732 1919 2147 1889 2077 2248 1954 2181 2564 1931 2108 2414

1741 1710 1726 1741 1710 1726 1697 1700 1714 1697 1700 1714

updates were implemented, but it should be noted that the updates did not detract from the ability of the CCK code to predict char behavior in conventional oxidation and combustion scenarios. Instead, it extended the submodels to also capture intense oxy-fuel conditions. Table 3 shows a summary of parallel experiments conducted with N2 diluent rather than CO2. These data were not used in the

Table 3. N2 Parallel Experiments Coal Black Thunder Pittsburgh

Utah Skyline North Antelope

O2 N2 Mol % Mol % 12 24 36 12 24 36 12 24 36 12 24 36

74 62 50 74 62 50 72 60 48 72 60 48

H2O Mol %

Peak Particle Temp (K)

Peak Gas Temp (K)

14 14 14 14 14 14 16 16 16 16 16 16

1861 2128 2289 1770 2154 2313 2091 2325 2520 2080 2357 2532

1677 1711 1753 1677 1711 1753 1690 1692 1712 1690 1692 1712

Figure 1. Logic map of CCK/oxy execution.

logic map of CCK/oxy execution, while Table 4 shows the main equations of the model. The model is currently configured for an entrained flow reactor (specifically the flat flame burner referenced in Section 2), so minor modifications may be necessary for other reactor types. 3.2. Previous Models. Several char conversion models include complex submodels that attempt to capture the most important chemistry and transport effects of char conversion. The code used here is an extension of the Carbon Conversion Kinetics (CCK) code10,20 with numerous additions to make the code functional and accurate in the extremes of oxy-coal combustion. These modifications include a more stable temperature solver with informed initial guess values that result in rapid convergence times, step-size independence, and successful model execution at extremely high temperature (appropriate for highly elevated O2 concentrations) or high H2O and CO2 concentration environments. A number of key submodels were also revised or replaced to more nearly approximate the physics of heterogeneous char conversion. Predecessors of the CCK code include Carbon Burnout Kinetics−extended (CBK/E)21 and Carbon Burnout Kinetics−gasification (CBK/G)22 codes (which grew out of the Carbon Burnout Kinetics or CBK code11). CBK/E utilizes a 3-step char oxidation reaction with O2 (reactions R1−R3) first introduced by Hurt and Calo,23 while

calibration of the CCK/oxy model, but they were collected under parallel conditions in the same apparatus, for the same coal, and with the same size cuts. It is of interest to determine whether kinetic parameters calibrated solely in conventional conditions can predict oxy-coal combustion data.

3. MODEL DEVELOPMENT 3.1. Model Summary. The CCK/oxy model is motivated by the inability of past comprehensive models to fit data taken in oxy-fuel conditions. This failure has been observed in previous work by Holland and Fletcher7 and by McConnell and Sutherland.8 While the present work was guided by a sensitivity analysis of the CCK model, that sensitivity analysis strongly implied that the most influential model parameters were the B

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Energy & Fuels Table 4. CCK/Oxy Submodels Submodel Name

Model Form

dTp

= hA p(Tg − Tp) + σεpA p(Ts4 − T p4) +

Particle Energy Balancea

mpCp

Surface Reactions10

2C + O2 → C(O)α + CO

dt

i

C(O)α → CO

C(O)β → CO

(R4)

(R5)

C + H2O ↔ C(O)γ + H2 C(O)γ → CO

(R8)

k1k 2PO22 + k1k 3PO2

R C − O2 =

k3 2

k1PO2 +

R C − CO2 =

R C − H2O =

(2) k4PCO2

1+

k4 P k5 CO2

+

k4r P k5 CO

1+

k4 P k5 CO2

+

k4r P k5 CO

dp

ρC υi(ni + 1)R i , s

2

2Deff , iCi , s

ϕi =

Effectiveness Factor

⎡ ⎛ ⎞⎤ 1 1 ⎟⎥ ηj = ⎢coth⎜⎜3ϕj − 3ϕj ⎟⎠⎥⎦ ϕj ⎢⎣ ⎝

Random Pore Model12

A p,0

1 − yi

(3)

+

k6 P k 7 H2O

+

k6r P k 7 H2

(4)

(7)

(8)

Di , j

= (1 − x) ×

2

1 − ψ ln(1 − x)

(9)

Low Heating Rate Particle Swelling13

⎛d⎞ = m log(Ṫ ) + b ⎜ ⎟ ⎝ d0 ⎠LHR

CO/CO2 ratio18,19

k6r P k 7 H2

yj

Species

∑ j = 1, j ≠ i

⎛ Ṫ ⎞ d = svar ⎜ base ⎟ d0 ⎝ Ṫ ⎠

Mode of Burning16,17

+

(6)

High Heating Rate Particle Swelling13

Devolatilization Model Gas Property Models14,15

k6 P k 7 H2O

(5)

Thiele Modulus

Ap

+

k 8PH2O

⎛ E ⎞ dfi = − fi Ad exp⎜⎜ A ⎟⎟ dt ⎝ RTp ⎠

Di , mix =

(R6)

(R7)

C + 2H2 → CH4

Multicomponent Diffusion

(R2)

(R3)

CO2 + C ↔ C(O)β + CO

Thermal Annealing11b

(1)

(R1)

C + O2 + C(O)α → C(O)α + CO2

Langmuir−Hinshelwood-type Reactions10

∑ rp,iΔHrxn,i

cHR

+ smin

(10) (11)

The CPD model is complex. Some detail is given below, and further details are referenced. Polynomials from tabulated data used to calculate gas phase thermal conductivity and heat capacity as a function of temperature and molar composition. dρp dmp η = dt dt Vp (12)

⎛ E ⎞ CO = Ac exp⎜⎜ c ⎟⎟ CO2 ⎝ RTp ⎠

(13)

a

All of the heat of reaction is applied to the char particle, and eq 13 dictates the ratio of CO to CO2. bThe annealing model is a complex set of statements to determine EA and Ad. Details are given below.

R7 represent distinct species with separate reactant pools, as denoted by the subscripts α, β, and γ. The above models successfully described the details of char conversion for oxidation, gasification, and pressurized gasification, but they were neither designed for, nor tested at the unusual gas compositions and combustion temperature found in oxy-coal

CBK/G introduced a 5-step gasification model with CO2, H2O, and H2 (R4−R5). CCK combined these two equation sets into a single, 8-step mechanism theoretically capable of handling the common gasification species (optionally at high pressure) within a single model.10 Note that in the 8-step mechanism shown in Table 4, the C(O) complexes in reactions R3, R5, and C

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ameliorated this issue, but there are still significant combinations of kinetic parameter space that lead to physically absurd results or outright model failure. It is impractical to search the entire kinetic parameter space as part of the optimization routine, but infeasible to put simple constraints on an irregularly shaped, high-dimensional kinetic parameter space. The details of the kinetics are discussed below, but the simple, practical solution was to explore the kinetic parameter space with a space-filling design. In this case, a Latin Hyper-cube design24 was used, and an alternative pathway detected and reported nonphysical parameter sets. This method was insufficient to wholly avoid physically infeasible space, but it did reveal the contours of several “valleys” of parameter space that contained local minima for an optimization objective function. By using these valleys as starting points for the final optimization, local minima were located with minimal intrusion into unphysical parameter space, and optimization routines became practical to execute. 3.4. Chemical Percolation Devolatilization Code. The chemical percolation devolatilization (CPD) model25,26 calculates the time-dependent release of volatiles as a function of coal type, heating rate, temperature, and pressure. The mechanism for thermal decomposition in the CPD model is directly related to the initial chemical structure, and the rates for cleavage of bonds between aromatic clusters are modeled. A Bethe lattice along with percolation lattice statistics is used to relate the number of cleaved bridges to the fraction of clusters that are not attached to the lattice. A flash calculation is used to relate vapor pressure to the amount of released tar vs the amount of Metaplast remaining in the particle. Cross-linking of Metaplast to the char particle is also modeled. The CPD code was recoded into a compatible format and linked to the CCK/oxy code to allow a complete prediction of the coal particle in a given set of circumstances. With a single set of inputs to describe the ambient environment, CCK/oxy produces a prediction that tracks the coal particle from raw coal, through initial heating and devolatilization, and throughout gasification and complete burnout. Because the thermal annealing kinetics are quite rapid, it is generally thought to be sufficient to execute the devolatilization code in the absence of annealing, and then to allow annealing to begin at the same time as combustion (despite most annealing actually taking place during the devolatilization phase). This is valid only if the consumption of char is negligible during devolatilization and if the annealing of the char “catches up” before significant amounts of char are converted. Both assumptions are reasonable in typical combustion regimes, but this assumption was tested here by intertwining the annealing with the devolatilization model and the char conversion model. No significant difference was manifested when allowing annealing and devolatilization to occur sequentially vs the more physically appropriate (but computationally onerous) concurrent computation of devolatilization and the annealing submodels. The CPD code and three other submodels employed in CCK/oxy (the kinetics, swelling, and annealing submodels) require information from the 13C nuclear magnetic resonance imaging (NMR) parameters. These parameters contain important structural information about the coal, but they are not generally available for most coals. Therefore, to maintain a high degree of applicability, the NMR parameter correlations reported by Genetti et al.27 were used. These nonlinear correlations were developed to predict the NMR parameters of any coal based only on the widely available proximate and

combustion. Further details of the CCK- and CBK-type models are available elsewhere.10,11,20−22 3.3. Practical Model Execution Considerations. The next several subsections outline the work done to improve the most influential submodels of the CCK/oxy code. While it is tempting (and often effective) to adjust some of the numerous uncertain parameters in the char combustion code until a desirable fit to data is obtained, this method of tuning to data results in a code that is often less predictive and less broadly applicable. Thus, with the goal of producing a combustion code with the widest possible applicability, the CCK/oxy model as a whole was not tuned to the specific data used here until the final optimization of the kinetic parameters. Instead, each submodel is in general compliance with char conversion theory and a subset of data related to the specific function of the submodel (i.e.; the swelling submodel was tuned to data that only related to swelling, etc.). The CCK/oxy model still contains several uncertain parameters related to such values as char tortuosity or ash grain size; since these values are generally unavailable for specific coals, default values are used. Prior to revising submodels, several practical issues had to be resolved in code execution. These same issues are quite likely to arise in any attempt to modify a char combustion code from conventional to oxy-coal conditions, and they are therefore worth mentioning briefly. First, oxy-coal systems tend to have a higher O 2 concentration to compensate for the slower diffusivity of O2 through CO2 and the cooling effects of endothermic gasification reactions. The high O2 concentrations can lead to very high local char combustion temperatures, and the high temperature, abundant O2, and concentrated CO2 and H2O combine for an ambient environment that converts solid carbon very rapidly. Because char conversion codes are typically numerical solutions of sequential time steps, the reactant surface partial pressure is used throughout the time step, and the especially intense conditions of oxy-fuel are not likely to be fully grid-converged in models that functioned well at conventional conditions. The second issue is a direct consequence of the first; unrealistically fast carbon conversion leads to excessive temperature spikes that diverge rapidly from experimental data. The solution to both of these issues was an adaptive time step tied to the particle temperature change. In the initial particle heating phase, when gasification and combustion reactions are negligible, the particle is permitted to change temperature significantly in a single time step, but if the magnitude of change is too great, then the step is retaken with a smaller time step. In the current model, 10 K in a single time step is found to be more than sufficiently restrictive to ensure grid convergence in the cold region. When the particle is hot enough to react at a meaningful rate, the time step is instead tied to the ability of the particle to rapidly converge to a new temperature and surface reactant concentration via diffusion and the particle energy balance (which leads to much smaller time steps). Together, the adaptive time step was effective in maintaining grid convergence and reasonably rapid model execution. The final practical code execution issue resulted from O2, CO2, and H2O all becoming important reactants in the oxy-coal environment. The balance between exothermic oxidation and endothermic gasification makes the energy balance difficult to converge even before attempting to match the model to data. More robust solver constraints and a relatively conservative and adaptive guess function for the new particle temperature D

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ultimate analysis, and they are described briefly below, with more detail available elsewhere.27,28 3.5. The Mode of Burning Parameter. Porous fuel particles are typically considered to gasify in one of three regimes or zones. They may be entirely kinetic limited, entirely film diffusion limited, or exhibit a mixture of internal diffusion and kinetic limitations. Kinetic limitations imply a relatively cool particle, which is not the case of coal char at practical combustion conditions. However, since gasification reactions with CO2 and H2O have enhanced importance in the oxy-fuel scenario, and because the associated activation energies are much higher than that of combustion in O2, kinetic limitations may well be expected to have a significant impact on char conversion. Because the three regimes have very different implications for char consumption, the concept of a mode of burning parameter, α, has historically been used to balance the shrinking diameter with the decrease in density in accordance with the char conversion regime (film diffusion limited, kinetically limited, or somewhere in between).29−31 An equation relating mode of burning, mass, and diameter is show below (eq 17), with α = 0 indicative of constant density and thus a complete film diffusion limitation, while α = 1 implies constant diameter with kinetic limitation. In conventional, air-fired char combustion, an α value of 0.2 is recommended.11,32 ⎛ m ⎞α ρ =⎜ ⎟ ρ0 ⎝ m0 ⎠

dt dρp dt

η≈

=

=

3 ϕ

dmp 1 − η dt 4πr p2ρp

(15)

dmp η dt Vp

(16)

(17)

In the case of a near-zero effectiveness factor, the particle is diffusion limited and the radius change represents the entire mass loss of the particle, while the density is essentially constant. In the case of η = 1, the char particle is kinetically limited, and the diameter is essentially constant. These equations are exactly true under the assumptions used in the derivation (first-order, irreversible kinetics modeled by the Thiele modulus at steady state, and a small effectiveness factor). In CCK/oxy, the effectiveness factors for reaction with O2, CO2, and H2O are computed at each time step, and then they are weighted according to the fraction of carbon consumption that is due to each reaction pathway. The weighted effectiveness factor for a given time step is used to compute the change in density for that time step, and the change in density is then used in conjunction with the computed conversion of carbon (from the kinetic and gas diffusion submodels) to compute the diameter decrease for the time step. This second step is done because, as mentioned above, some of the assumptions in the derivation of eqs 15 and 16 are only approximations in the reality of char combustion. Thus, eq 15 for the change in radius is superseded by enforcing the law of conservation of mass, which corrects for the inconsistency introduced by the approximations of the derivation. 3.6. Coal Particle Swelling Model. The swelling submodel employed in the CCK model is overly simplistic. The most advanced swelling models attempt to capture the physics of bubble formation,33−35 but are impractical because they are computationally expensive, complicated to implement, and may not be applicable to a wide range of coals. Most importantly, until recently,34,35 these swelling models do not follow the observed swelling trends at the extremely high heating rates (104−106 K/s) relevant to practical coal combustion systems.36 This is true of all swelling models that neglect the functional dependence on heating rate, and is presumably because the extreme heating rates drive the volatiles from the particle very rapidly.21,37 The rapid loss of volatiles leads to a very short time frame for bubble formation, and when the rate of bubble growth exceeds the rate of Metaplast relaxation, the bubbles “pop”, leading to an entirely different swelling regime.38 The swelling model implemented in CCK/oxy incorporates information about the coal structure and type as well as heating rate dependence, with the structural parameters predicted from the NMR correlations mentioned previously. Both the coal type and heating rate heavily impact the swelling behavior.37 The newly implemental model was developed by Shurtz et al.,13,20 and a brief description of equations and applicability is given below, with details of the swelling model development given elsewhere.13,20 The swelling ratio is given by eq 18, where svar, cHR, and smin are described by the correlations in Table 5, and Ṫ is the maximum heating rate that the particle experiences during the heat-up and swelling process (in K/s, as estimated via the transient particle energy balance). This maximum rate

(14)

In the construction of the CCK model, it was intended that the model should be run at either combustion conditions or gasification conditions, while oxy-coal combines the high temperature and O2 conditions of combustion with the high CO2 (and sometimes H2O) concentration of gasification. Thus, while the CCK code could match data well with a simple heuristic to determine α, the CCK/oxy model is a significantly more complicated situation. Moreover, oxy-coal might have a very wide range of O2 concentrations, depending on the specific application, which in turn leads to a broad range for particle temperature that heavily influences the value of α relevant to gasification. Finally, global coal combustion models (and CCK) simply designate a mode of burning as a constant for a given instance, but in reality the balance between diameter loss and density decrease is far from static during char particle burnout. Given the sensitivity of the CCK model to α,7 it was necessary to significantly improve the mode of burning implementation in CCK/oxy using a variation on a method derived and applied by Haugen et al.16,17 This method is derived in detail elsewhere,17 but the key results are shown in eqs 15 and 16. The results below were obtained using the Thiele modulus relevant to a first-order, irreversible reaction at steady state, and also assumed a relatively large value of the Thiele modulus such that the effectiveness factor could be approximated by eq 17. In reality, the order of the combustion reaction is a matter of controversy that depends both on how the kinetic system of equations is framed and on the temperature regime; also the effectiveness factor may not be small for gasification reactions. However, the method shown in the equations below is considerably superior to a single, fixed value of α, and establishes a conceptually sound relation between the changes in particle diameter and density for the reactive regime of the oxidative gases. E

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in a thermogravimetric analysis (TGA) device, the total volatile yield may be considerably less than if the same particle is heated at 105 K/s in a flat flame burner. The volatiles that are not released at low heating rate instead cross-link back into the coal structure, giving the char particle a different chemistry, and altering not only the rate of annealing, but also the annealing pathways available based on the heating rate of char preparation. Alternatively, annealing experiments may commonly heat a char particle rapidly to approximately 1300 K. This temperature is often insufficient to melt the inorganic ash in a char particle, and molten inorganic ash loses all catalytic activity, while inorganic crystals may have catalytic activity for both carbon conversion and carbon crystal rearrangement reactions; thus, peak particle temperature can also impact both the extent of annealing and the available annealing pathways. In the equations below, f is the fraction of active sites in bin i, k is the preexponential factor for the rate of active site loss (constant for all bins), and EA,i is the activation energy from the ith bin of the log−normal distributed activation energy profile. The parameters a−d were optimized from a large collection of data tabulated elsewhere, while σ and μ are parameters for the log−normal distribution, and p0 is the fraction of intact bridges from the NMR structural parameter correlations. Finally, there are also two truncation factors that transform the log−normal activation energy distribution into a bimodal distribution to represent the two main sets of annealing reaction pathways (devolatilization and thermal deactivation). Parameter values for this work are shown in Table 6.

Table 5. High Heating Rate Swelling Model Parameter Correlations Correlation

svar =

σ+1 1.69 M δ

svar =

σ+1 − 3.37 M δ

Applicable Range

− 0.0309 + 1.01

σ+1 Mδ

< 0.207

0.207 ≤

σ+1 Mδ

≤ 0.301

σ+1 Mδ

Svar = 0

cHR = − 191

0.018 ≤

2

( ) σ+1 Mδ

+ 68.9

σ+1 Mδ

− 5.16

CHR = 0

< 0.018 or

0.106 < σ+1 Mδ

σ+1 Mδ

σ+1 Mδ

> 0.301

< 0.254

< 0.106 or

σ+1 Mδ

> 0.254

occurs at initial heating, when the cold particle experiences the greatest temperature gradient with its surroundings. Table 5 (reproduced from Shurtz et al.13) gives the values for other variables of interest, as well as their range of applicability. Here σ+1 indicates the coordination number, Mδ refers to the average molecular weight of the side chains, HHR applies to the high heating rate regime, and FCASTM and AASTM are the American Society for Testing and Materials (ASTM) values for ash and fixed carbon content, respectively. Ṫ base is set at 5.8 × 104 K/s. ⎛d⎞ ⎛ Ṫ ⎞cHR = svar ⎜ base ⎟ + smin ⎜ ⎟ ⎝ Ṫ ⎠ ⎝ d0 ⎠ HHR

(18)

smin = (FCASTM + AASTM )1/3

(19)

The preceding equations and parameters introduce the vital elements of heating rate and coal structure into the coal particle swelling model. Coal structure in particular is introduced via correlations with the NMR parameters mentioned previously, and it allows for superior correlation than the less informative parameters of the proximate and ultimate analysis used previously. Equations 18 and 19 are intended for maximum heating rates of at least 8.3 × 103 K/s, and they have been shown to fit data taken at relevant heating rates and atmospheric pressure.13 The CCK/oxy implementation of the swelling model also incorporates a plugin for adding in the influence of high-pressure on swelling (also developed by Shurtz et al. and detailed elsewhere10). For lower heating rates, Shurtz et al.13 developed a piecewise correlation. The details are not relevant to typical combustion modeling, so, while CCK/ oxy contains the low heating rate correlation, the details are not included here. 3.7. Thermal Annealing. The thermal annealing model encompasses the changes in raw coal due to both preparation conditions and activity loss during burnout. The initial chemical and physical changes during devolatilization are by far the most impactful, but in the intense heating conditions typical of industrial coal combustion, both sources of thermally induced changes to coal reactivity may occur on roughly the same timescale, making it impossible to effectively separate the two effects. An extended annealing model that addresses both annealing effects is developed and discussed in detail elsewhere. 9 The results of the annealing model were incorporated in this work in eqs 20−25. Briefly, this annealing model differs from previous models used in comprehensive char conversion codes in that it accounts for both the rates of various annealing kinetic pathways and the changes to those pathways due to coal type, heating rate, and peak particle temperature. For example, if a given coal is heated at 10 K/min

Table 6. Annealing Parameter Values Parameter

Value

k0 a b c d

1.398 × 1012 s−1 0.356 ln (kcal/mol) 3.65 × 10−4 ln (kcal/mol) 1.531 ln (kcal/mol) 0.679 ln (kcal/mol)

The variable f merits slightly more explanation. Because the total number of active sites is unknowable, the total number of sites assigned to a given bin is equally unknowable. Fortunately, the annealing model is only needed to calculate the rate of active site loss relative to the original reactivity of the particle, so the number of sites in each bin may be normalized by the unknown original number of sites. Thus, in theory, f i = Ni/N0 (where N is the number of active sites), but in practice f i is some value between 0 and 1, and the values of all f ’s (initially) sum to 1. Once the number of bins is set, the initial values for f i are set to conform to a probability density function (PDF), and in this case a discretized, truncated log−normal PDF was found to be appropriate. The sum of the f i’s at any time becomes the fraction of sites that are active, and hence becomes a multiplier in the reaction rate expression that starts at 1.0 and decreases with time. ⎛ −E Aanneal , i ⎞ = −Ad exp⎜ ⎟f dt ⎝ RT ⎠ i

dfi

PDF(E Aanneal , i) =

(20)

2⎞ ⎛ 1 ⎛ ln(E Aanneal , i /μ) ⎞ ⎟ ⎟⎟ exp⎜⎜ − ⎜⎜ σ E Aanneal , iσ ⎠ ⎟⎠ ⎝ 2⎝

1

(21) F

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Energy & Fuels μ = ap0 + bTpeak + c

σ=

Ad =

Ad =

d p0

the data are not actually “noisy” in the traditional sense. Instead, the data indicate the actual temperature variation due to particle-to-particle variation in ash content and/or maceral character (variation due to measurement uncertainty may be on the order of ±25 K).32,39 4.1. Particle Diameter Determination. As shown in the Experimental Section, the data are for four different coals burned at each of three conditions. The data collection system was able to measure the diameter and temperature of individual particles at predetermined heights in the burner. Table 7 and

(22)

(23)

p0 k 0 ln(104)

for HR ≥ 104

p0 k 0 ln(HR + 2.7)

for HR < 104

(24)

(25)

Table 7. Utah Skyline Data Summary

3.8. Gasification and Oxidation Kinetic Equations and Parameters. The reaction steps R1−R8 each have an associated activation energy and preexponential factor for a total of 20 kinetic parameters (including two reverse reactions). This kinetic scheme was given in the CCK model as a combination of the combustion kinetic scheme from CBK/E and the 5 gasification reactions from CBK/G. A system of only eight reactions is an extremely simplistic skeletal mechanism attempting to capture the most prominent collective effects of innumerable unknown coal combustion reactions, but it has nevertheless proven sufficiently flexible to fit a broad sampling of combustion and gasification data.10,22 In fact, the kinetic system generally has more than enough free parameters, and the data of a given combustion scenario are typically insufficient to effectively optimize 20 parameters. This was recognized in the introduction of CBK/G, and it is typically sufficient to fit only the four kinetic parameters involved in R3 and R7 (k3, E3, k7, and E7), where R3 is the principle combustion reaction and R7 is the principle gasification reaction. The rest of the kinetic parameters are either fixed at nominal values or tied to the kinetic parameters of R3 and R7 via correlations developed with CBK/G.22 After the preceding submodels were updated or added to the CCK/oxy model, the kinetic parameters were optimized. It is important to note that this optimization is not a fit of the kinetic parameters as is typically observed with a single, linearized reaction equation fit to a set of rate data. Instead, the entire parameter space is given bounds and, optionally, both linear and nonlinear constraints. An initial guess value was provided for each parameter, and the optimization algorithm fmincon (from the MATLAB Optimization Toolbox) explored the constrained parameter space to minimize the error of an objective function.

Height (cm) Avg Dp (μm) Avg Tp (K) Number of Points Height (cm) Avg Dp (μm) Avg Tp (K) Number of Points Height (cm) Avg Dp (μm) Avg Tp (K) Number of Points

12% O2 7.62 8.89 106 102 1864 1857 231 184 24% 5.08 6.35 96 97 2092 2088 195 233 36% 6.35 7.62 111 120 2290 2272 859 295

10.16 96 1810 1078 O2 7.62 102 2071 214 O2 8.89 119 2256 201

12.70 98 1815 198 10.16 103 2049 180

11.43 106 2038 195

12.70 107 2013 100

10.16 124 2239 162

11.43 123 2219 75

12.70 122 2180 29

Table 8 show the mean particle diameter and temperature for each height for two of the coals, but the other two coals reported only the average temperature data, which are shown in Table 9 and Table 10. The more complete data for Utah Skyline and North Antelope coals have several implications for the behavior of the experiment. Most importantly, the cohort of particles observed at a given height is NOT the same cohort as observed at lower burner heights. The burner height is adjusted between experiments, so the individual particles are different specific particles, but more importantly, the average characteristics of an observed particle change based on observation height. This is because of at least four competing effects that alter which particle cohort is most likely to be “seen” by the detector system: 1. The detection method relies on the light emitted by burning particles, so some particles simply are not detected. A particle is most likely to be detected if it is large and hot (and thus emitting a relatively large quantity of light). 2. Small particles tend to burn hotter than larger particles, but have less surface area and thus emit less total light than a larger particle of the same temperature. 3. As particles burn, the smaller particles reach near extinction relatively quickly, and without the heat of reaction, the particle temperature drops below the gas temperature due to radiative losses. These nonburning or slow-burning particles then become invisible to the detection system. This increases the average diameter of the detected particles and decreases the number of particles detected. This effect is especially important for the very small particles that are below the nominal size cut (either from fragmentation or from imperfect sieving).

4. RESULTS AND DISCUSSION Model development essentially consisted of two steps. First, numerical and model execution issues from CCK that caused the CCK/oxy model to fail in the extremes of the oxy-coal environment were eliminated. Second, the most sensitive submodels were replaced with updated, more physically realistic submodels. Because the models are simply more refined, rather than specific to the oxy-coal environment, improvements realized by the updated model are valid in gasification, airfired, and oxy-coal conditions. This section compares the relatively limited oxy-coal data to the CCK/oxy model, after optimizing the kinetic parameters as described in Section 3.8. In considering the results, note that the data have a very wide range of particle temperature values; at any given observation height and particle diameter, the particle temperatures range on the order of ±150 K from the mean. However, the trends in the data may be due to more than merely noisy data, and in fact, G

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Energy & Fuels Table 8. North Antelope Data Summary Height (cm) Avg Dp (μm) Avg Tp (K) Number of Points

7.62 86 1851 176

Height (cm) Avg Dp (μm) Avg Tp (K) Number of Points

5.08 92 2038 191

Height (cm) Avg Dp (μm) Avg Tp (K) Number of Points

5.08 100 2294 200

8.89 89 1853 149 24% O2 6.35 96 2054 211 36% O2 6.35 106 2323 200

10.16 88 1873 131

11.43 88 1867 151

12% O2 12.70 91 1870 100

7.62 96 2059 293

8.89 97 2077 129

10.16 100 2093 37

7.62 110 2348 157

8.89 96 2363 105

10.16 92 2370 23

13.97 86 1859 59

15.24 96 1876 54

16.51 96 1854 52

17.78 94 1860 31

19.05 103 1826 19

Table 9. Black Thunder Data Summary Height (cm) Avg. Tp (K) 24% O2 Height (cm) Avg Tp (K)

5.08 1703

5.715 1708

6.35 1726

4.45 1837

5.08 1894

Height (cm) Avg Tp (K)

3.18 2038

3.81 2052

5.715 1913 36% O2 4.45 2103

12% O2 7.62 1732

10.16 1725

15.24 1715

20.32 1718

6.35 1914

7.62 1919

10.16 1879

12.70 1859

5.08 2147

7.62 2135

10.16 2072

25.40 1690

Table 10. Pittsburgh 8 Data Summary Height (cm) Avg Tp (K)

5.08 1822

5.715 1824

Height (cm) Avg. Tp (K)

5.08 2037

5.715 2006

12% O2 6.35 1872 24% O2 6.35 2054

Height (cm) Avg. Tp (K)

3.18 2028

3.81 2179

4.45 2207

7.00 1855

7.62 1873

8.89 1889

7.62 2066 36% O2 5.08 2242

8.89 2077

10.16 2067

5.715 2248

6.35 2237

10.16 1898

7.62 2245

8.89 2238

10.16 2186

height. In general, the profiles of initial particle diameter and observation height (shown in Table 11) are taken from the diameter profiles of the observed data (in the case of North Antelope and Utah Skyline), or extrapolated from the North Antelope and Utah Skyline diameter profiles (in the case of Black Thunder and Pittsburgh 8). The extrapolation of North Antelope and Utah Skyline diameter profiles onto Black Thunder and Pittsburgh 8 (respectively) is necessary because the diameter profiles are not reported, but it is done on the frankly tenuous grounds that Black Thunder and North Antelope coals are both subbituminous while Pittsburgh 8 and Utah Skyline are both high-volatile bituminous coals. The resulting model/data fits are surprisingly good, but it must be emphasized that the Pittsburgh 8 and Black Thunder diameter profiles are extrapolations. It should also be emphasized that the diameter profiles are determined from the available data, and then the data are fit by adjusting only the kinetic parameters. Thus, the coal particle diameter is somewhat uncertain, but it is NOT a free parameter, and it is not adjusted to get the “correct” fit. The observed diameters during late-burnout are smaller than their initial value because the particles have decreased

4. Large particles take longer to burn out into the undetectable range because they have both greater mass and a larger surface area to emit light. However, these initially large particles persist into later residence times and decrease in diameter during burnout, which decreases the average diameter of the detected particles. Bearing the above effects in mind, it is not easy to immediately apply simple trends to the data, but some approximations are certainly necessary to input a more accurate particle size into the CCK/oxy model so that the comparison between data and model is legitimate. One obvious trend is the change in observed particle diameter between conditions. In general, more intense conditions have a larger average observed particle diameter, most likely because the high O2 concentration rapidly consumes the smallest particles and leaves a higher fraction of larger particles. Failure to account for these trends in coal particle diameter with burner height and O2 condition leads to the inability of a model to match all of the data (i.e., if the model fits for the 12% O2 environment, it will not fit for the 36% and vice versa). The data clearly necessitate that different initial diameters be allowed for comparison with data from each measurement H

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Energy & Fuels Table 11. Multiple Diameter Profiles Coal Type O2 %

Black Thunder 12

Height (cm) 3.18 3.81 4.45 5.08 5.72 6.35 7.00 7.62 8.89 10.16 11.43 12.70 13.97 15.24 16.51 17.78 19.05 20.32 25.40

24

North Antelope 36

12

Diameter (μm)

90 90 90

90 90 90 90

90 95

115 105

Pittsburgh 8 36

12

Diameter(μm) 90 90 90 90

105 105

24

115

86 89 87 88 91 85 96 98 97 107

24

Utah Skyline 36

12

Diameter (μm)

92

100

96

106

96 101 106

110 111 106

98 98 98 98 98 98 98

24

36

Diameter (μm)

95 98 98

95 95 100 105 110 110

100 100 105

120 120 125

96

106 103 101 103

97

111

102

120 119 124 127 128

107 112 115

105 115

somewhat in size during burnout. The CCK/oxy model accepts the initial diameter as an input and then predicts the diameter, temperature, and mass change of the particle throughout burnout. Therefore, where there is evidence that the observed diameter has decreased from the initial value (i.e., at later burnout times), this diameter change should be taken into account. The initial diameter is estimated from the observed diameter when modeling the late burnout data. It should be noted, however, that while the profiles are reasonable given the size cut, predicted swelling, and diameter loss due to burn out, the late-burnout diameters are a slight extrapolation. The specific profiles of each coal have a few points worth noting: 1. The extrapolated profile for Black Thunder and Pittsburgh 8 were simply set at a lower bound of 90 and 95 μm, respectively. Those coals were observed at much lower heights than the North Antelope and Utah Skyline coals, but, given the particle size cut, it is unreasonable that the average particle diameter would continue to follow a sharp downward trend at low observation heights. 2. In general, the North Antelope profiles are directly from the data with small increases of between 2 and 6 μm added to the initial diameter for data obtained at locations after the diameter data are observed to “peak”. This peak indicates that the largest mean diameter has been reached from the effect of smaller particles burning out. Following the peak, the diameter decreases as diameter loss due to char conversion becomes the more prominent effect. This causes the mean diameter to gradually decrease or not increase as sharply as would otherwise be expected, and to get a closer estimate of the mean initial diameter for the observed late-burnout particles, it is necessary to estimate the diameter loss. In the last two points of the 36% O2 condition, the mean particle size dropped quite substantially because of the rapid burnout of the intense condition. Here, 15 μm was added, to bring the value approximately to the peak observed size; given that, generally, the largest particles

will survive to late burnout, it is thought that the peak value is as good an estimate as can be made for late burnout particles. 3. The Pittsburgh 8 coal diameters are a rounded extrapolation from Utah Skyline diameter data. Also, the 12% O2 condition in the Utah Skyline is too narrow and noisy to be overly informative, so there was no justification for extrapolating a particular profile to Pittsburgh 8. Instead, the Pittsburgh 8 12% O2 input was fixed at a rounded mean of the extrapolation (98 μm). 4. The Utah Skyline coal 12% O2 environment is an anomaly in that the data begin with a relatively high diameter and then exhibit marked decreases in both diameter and particle temperature. Upon close examination of the complete set of relevant data (rather than just average values), it seems likely that fragmentation is the cause of this unusual behavior. The first two observation heights in the 12% O2 environment have a relatively high proportion of large particles. However, at the third observation height (10.16 cm), the data show a significant increase in very small particles when the opposite trend is expected and observed in the remaining data. Specifically, the second observation height has nearly six times as large a proportion of very large (>180 μm) particles over the third observation height. Additionally, the second observation height has a 7-fold higher proportion of relatively hot particles (>1900 K), while the third observation shows a substantial increase of small, cold particles and a 5-fold increase in total particles observed. 5. Like North Antelope, the Utah Skyline particle diameter in 36% O2 is observed to peak and then decrease, so the original diameter of the cohort is crudely estimated to be between 4 and 8 μm higher than the observed diameter, based on the amount of postpeak decrease and the range of observed diameters. I

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Figure 2. Comparison of CCK/oxy model calculations with coal data from Shaddix and co-workers4,5 using the measured particle diameters.

4.2. Model/Data Fit. The results of fitting the revised CCK/oxy model to the Shaddix data are shown in Figure 2 and Table 12. In this figure, the kinetic parameters are constant for

very low observation heights for the Pittsburgh 8 data appears to be incorrect. In the extrapolation, the 36% O2 Pittsburgh 8 data were assigned diameters as low as 95 μm, because the observations began much earlier than Utah Skyline (where the lowest diameter was observed to be 111 μm). By changing the Pittsburgh 8 early burnout diameter values to 105 μm (still inline with the known early burnout values at the Utah Skyline 36% O2 condition), the mean absolute error was reduced to 36 K and the maximum error decreased to 65 K even without reoptimization. The sum squared error also decreased by more than 50% (also without reoptimization), because the single point in question contained more error than the sum of the entire data sets at all three O2 conditions. Because of the dominant weight of that one point introducing a skew in the optimization objective function, reoptimization removed the skew and reduced the mean and maximum absolute errors in the 36% O2 Pittsburgh 8 data to 21 and 42 K, respectively, effectively eliminating one of the two remaining sources of notable bias. This is not shown in the plot, and is not reflected in the error tables because the updated result depends on an uncertain inference based on postoptimization analysis. It seems likely that the extrapolation of the Pittsburgh 8 early burnout 36% O2 condition was a few microns off, because a small change in the extrapolation makes all of the Pittsburgh 8 data self-consistent, but the actual cause of the error is unknown. The second largest maximum error coincides, as expected, with the other noteworthy bias. The 24% O2 environment for Utah Skyline coal has a bias from one of several possible sources: bias in model prediction due to imperfect submodels, inappropriate extrapolation for the initial diameter inputs, a skew in the data, etc. The extrapolation of the diameter data seems the mostly likely culprit, and it must be emphasized that all of the diameter profiles are, to some degree, an extrapolation. However, the extrapolation is quite reasonable to within a few microns given the data trends, and since the

Table 12. Difference between Calculations with Multiple Diameter Profile and Measured Particle Temperatures Black Thunder

Mean Absolute Error (K)

Max Error (K)

Pittsburgh 8

14 26 40 Max Error (K)

12% O2 24% O2 36% O2

North Antelope

9 16 18 Mean Absolute Error (K)

12% O2 24% O2 36% O2

13 28 13

43 52 27

12% O2 24% O2 36% O2

Mean Absolute Error (K)

Max Error (K)

Utah Skyline

11 9 49 Mean Absolute Error (K)

22 37 157 Max Error (K)

12% O2 24% O2 36% O2

33 41 17

48 81 44

any given coal, and the initial particle diameter is input into the CCK/oxy model. The model then predicts a temperature vs height profile, which is plotted from initial heat up (starting at 300 K) but truncated before complete burnout at the relevant burner height. The truncation improves graph readability, but it should be noted that the model appropriately predicts a late burnout drop in temperature. The mean absolute error values are on the order of the noise due to measurement error (±25 K). Systematic bias is nearly absent, and maximum errors fall in the expected range of measurement error, with two notable exceptions. First, the Pittsburgh 8 coal has an enormous maximum error (see Table 12) that is introducing a skew into the entire data set, especially in the 36% O2 environment (note that, though the error is large, a diameter change of only a few microns has a very profound impact on the earliest observed particles because they are still in a very rapid heating phase). The error is because the extrapolation of coal diameter to the J

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at a given height. The diameters required to match the temperature data are uniformly unfeasible for such late burnout given the rest of the data, but this is in fact exactly what would be expected. The particles that survive to late burnout are either exceptionally persistent or they are in a near-extinction regime or near-burnout as discussed by Sun and Hurt.19 In the case of the persistent particle, they are likely larger particles that have a particular ash content and/or maceral character that requires a longer residence time to consume. In the case of nearextinction or burnout, the particles have experienced a very significant and rapid transition to a nearly inert particle that is heated almost entirely by ambient conditions (rather than exothermic oxidation). CCK/oxy predicts both states, and it is seen in the rapid decrease of the temperature profile (corresponding to roughly the last 15% of burnout), followed by a long, slow decline in particle temperature. (The slow decline is due largely to the continual decrease in ambient gas temperature with height in the particular experimental setup.) Because the transition between burning and near-extinction is quite rapid, and the particles are far from uniform in their burning characteristics, it is quite unlikely that the observation height would happen to be appropriate to observe a significant number of particles midtransition. Instead, the observations are an average of particles that are just barely hot enough to be detected, and particles that continue to burn. This average is weighted by the proportion of particles that are near burnout vs those still burning rapidly, and that weight shifts (as expected) toward near-burnout for progressively longer residence times. Also as expected, this shift is more rapid for more intense O2 environments. In short, the Pittsburgh 8 late burnout data support the validity of the kinetic optimization done on earlier burnout data, and all but one of the particles fall within an appropriate diameter window (as determined by the reference diameter sizes) and show a reasonable weighted average. The largest disagreement between the model and the data in Figure 3 is the last point in the 36% O2 environment. No diameter input into CCK/oxy, no matter how small, intersects that point, indicating that it is essentially completely converted, and the relevant energy balance is that of an inert particle. This in turn means that the kinetic submodel (and all other submodels) in CCK/oxy is no longer relevant to that particle. Instead, only the energy balance is relevant, and no set of coalspecific inputs into CCK/oxy allow it to intersect the point in question, which leaves two potential conclusions: (1) Some of the standard energy balance assumptions are incorrect in this case (true, but probably not significant), or (2) the gas temperature profile and/or environmental wall temperature profile are slightly incorrect (also true in some degree, and more significant). 4.4. Kinetic Parameter Results. The optimized values of the kinetic parameters for this data set are given in Table 14.

profiles were set prior to optimization, not adjusted postoptimization to match the CCK/oxy model, it is likely that any set of parameters following the observed trends in the data would be able to obtain an excellent fit with slightly different kinetic parameters. Thus, the data trends are important, but the actual diameter values only need to be correct to within a few microns. This last observation highlights a potential difficulty in applying the CCK/oxy model. Frequently, coal particle burnout is determined by collecting a sample of partially burned char and measuring the level of conversion. Such an approach does not suffer from the weakness of observing only progressively more truncated portions of the total particle feed. However, data such as those used in the present work represent only a portion of an unknown feed distribution. Thus, while the predictions and data show excellent agreement, and the CCK/ oxy model is expected to perform well for the entire particle distribution, only a portion of the distribution is available for direct comparison and validation. A solution to this problem for computational fluid dynamics (CFD) applications is addressed in greater detail in Section 5, but in general, the data here imply that nearly the entire particle distribution is “seen” in the 12% O2 case, and hence one form of validation might be to extrapolate from the 12% O2 case. 4.3. Late Burnout. It should be noted that the Pittsburgh 8 data also included a selection of data points at far higher observation heights than any of the other data. Predicting the relevant diameters would have been an extreme extrapolation, so they were excluded from the optimization, but they are shown below in Figure 3 and Table 13 (a discussion of relevant kinetics follows in the next section).

Figure 3. Comparison of CCK/oxy model calculations with late burnout Pittsburgh 8 coal data from Shaddix and co-workers4,5 using the measured particle diameters.

Table 13. Diameter Values That Match Late Burnout Predictions of Particle Temperature Diameter Designation Lower Diameter Diameter Match Diameter Match Diameter Match Upper Diameter

Bound (μm) (μm) (μm) (μm) Bound (μm)

12% O2

24% O2

36% O2

60 77 89 100 125

60 93 104

60 91

120

125

Table 14. Kinetic Parameters Optimizations Parameter k3 (mol/cm3) k7 (mol/cm3) ER,3 (kJ/mol) ER,7 (kJ/mol)

Initial diameter values were selected such that CCK/oxy to precisely predict the measured mean particle temperature (upper and lower particle diameter bounds were also plotted for reference). These diameters were not selected on any basis other than that they fit the measured mean particle temperature K

Black Thunder

North Antelope

Pittsburgh 8

Utah Skyline

2.18 × 1010

1.39 × 1010

2.13 × 108

2.39 × 1010

1.21 × 109

1.60 × 109

3.63 × 108

7.28 × 109

117

117

180

130

210

240

267

272

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Energy & Fuels Table 15. Conversion Fraction due to Each Reactive Gas (particle diameter of 100 μm) Black Thunder

North Antelope

Pittsburgh 8

Utah Skyline

O2 Condition

O2

CO2

H2O

O2

CO2

H2O

O2

CO2

H2O

O2

CO2

H2O

12% 24% 36%

0.72 0.79 0.83

0.25 0.18 0.15

0.03 0.03 0.03

0.82 0.85 0.86

0.16 0.13 0.11

0.02 0.02 0.02

0.89 0.87 0.87

0.09 0.11 0.11

0.02 0.02 0.03

0.82 0.82 0.85

0.15 0.14 0.12

0.03 0.03 0.04

optimizations were performed using the same diameter profiles shown in Table 11, but only the four data sets from the 12% O2 environment were used. Then, the resultant kinetic parameters (Table 16) were used to extrapolate to the 24% and 36% O2

These values represent a local minimum in the kinetic parameter space, and are not unique values. This is expected and in fact unavoidable in any skeletal reaction mechanism because each reaction (R1−R8) represents an umbrella reaction for an enormous number of similar reactions involving the complex carbon chemistry of the char, and the kinetic parameters therefore have little physical meaning. Even a perfectly correct, physically meaningful model will have an infinite number of feasible parameter sets located in a “valley” in parameter space because of the noise in the experimental data. In the case of highly autocorrelated parameters (such as A and E in the Arrhenius form), the “valley” in parameter space takes on a distinctive shape, and in the case of high dimensional parameter space in a model with limited physical meaning and correlated parameters, the single “valley” becomes many, exceptionally narrow, “valleys”, each with a local minimum. This is the case of all but the simplest kinetic schemes, and there is no analytical solution to find either a local or absolute minimum, so optimization algorithms are used instead to find a local minimum. The location of the local minimum will depend on the initial guess value, but as long as appropriate constraints are set, the minimum of one “valley” is as valid as another. The optimization routine was executed from several different initial guess values and generally found an equivalent “valley.” In the small fraction of cases where the optimization ended in a substantially different local minima, failure was obvious from the results of the objective function, and the failed optimization was discarded. The kinetic parameters in Table 14 apportion carbon consumptions between the three reactive gases (O2, CO2, and H2O) as seen in Table 15. There are two trends of interest: first, with the exception of Pittsburgh 8, the consumption due to O2 increases with higher O2 concentration, which is unsurprising. Second, and somewhat surprising, the very high levels of O2 do not completely marginalize the conversion due to CO2. This is likely due to the relative magnitudes of the temperature-dependent exponential term in the relevant gasification and combustion Arrhenius equations. The high activation energy of the gasification reactions makes the exponential terms in the rate equation increase more rapidly with temperature than for the combustion reaction, and hence the gasification reactions stay significant at the high temperatures reached by the high O2 concentrations. 4.5. Extrapolation from 12% O2 Data. The literature data used here are of exceptional quality and detail, with a very wide range of O2 concentrations and parallel experiments using both CO2 and N2 as the gas diluent. Unfortunately, this level of detail and O2 range is unusual, so it is desirable to determine whether or not the CCK/oxy model is effective when extrapolating from only a single O2 concentration, rather than the entire span of the O2 range. The 12% case was chosen for the single condition optimization because, while it is the least informative about the effects of extreme conditions, the overwhelmingly most common literature scenario is a relatively low O2 concentration. The following kinetic parameter

Table 16. Kinetic Parameters of the 12% O2 Oxy-Coal System Parameter

Black Thunder

North Antelope

Pittsburgh 8

Utah Skyline

k3 (mol/cm3) k7 (mol/cm3) ER,3 (kJ/mol)

5.06 × 108 6.37 × 108 178

1.00 × 108 4.71 × 108 180

1.90 × 108 4.23 × 109 8 180

6.54 × 107 1.16 × 108 180

conditions for their respective coals. The results are shown in Figure 4 and Table 17, while the bullet points below highlight several important points of the results: 1. Only the activation energy and preexponential factor of R3 were optimized in this case. This was done because a brief exploration of the gasification parameters showed that any reasonable set of gasification kinetic parameter values gave equivalent results both at the 12% O2 gas composition and at the higher concentrations. It was observed that the optimization routine provided no significant inference regarding the gasification parameter values, because the gradient in the gasification dimensions of parameter space was close to zero for the 12% O2 data (i.e., the gasification reaction was weak relative to the noise at low temperature conditions). While the gasification kinetic parameters are certainly important for the 24 and 36% O2 conditions, little could be inferred about them from the 12% optimization, so they were fixed at the values found in the previous optimization. 2. Because optimizations in complex parameter spaces very often only find local minima, each kinetic parameter optimization was executed four times with significantly different initial guess vectors. The resultant values were sometimes quite different in individual optimizations of k3 and E3 (as expected in a complex space of many “peaks” and “valleys”), but the total rate constant and the goodness of fit (from the sum squared of the error of the objective function) were in excellent agreement. The figures below were generated using one of the four sets of results (chosen at random), and the values in Table 17 are averages of the replicate results. 3. In evaluating the usefulness of the extrapolations shown below, two distinct questions must be asked, and this evaluation endeavors to answer only one of the two. First, it is of interest to know if the extrapolated curves have the correct shape and magnitude. In this case, that means: (1) are the particle predictions roughly the correct temperature, and (2) does the shape of the prediction curve follow the shape of the data. The second question of interest determines how well the time axis L

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Figure 4. Predictions in oxy-coal conditions from 12% O2 data only.

The small variation on the time axis is entirely within the variation of particle size and character, and the resulting large temperature change from such a small variation emphasizes the impact and importance of describing the char particles as a distribution rather than a point estimate. To accurately answer the question of prediction trends and magnitude, the first point of the Black Thunder coal and the last point in the Utah Skyline coal from the 36% O2 environment are not included in the average error values in Table 17. 5. The North Antelope and Black Thunder temperature extrapolations were observed to consistently underpredict the data by a small amount. This is due to a slight imbalance between the gasification and combustion carbon conversion pathways. However, as described in point 1 above, other reasonable values of the gasification kinetics neither exacerbate nor correct this deficiency, so it is thought that the noise of the 12% O2 data can be better accommodated with slightly less aggressive combustion kinetics than the value suggested by the entire body of data. These less aggressive kinetics have a very small positive impact on the 12% data under optimization simply because of random chance in a small data sample. In the extrapolation to the entire data set, this translates to a small negative impact on the rest of the data set. No immediate solution presents itself; small, uniform data samples are simply of less value that larger, more diverse samples. 6. On the whole, the results below show that the extrapolation from low O2 concentration up to very extreme concentrations was remarkably successful, and the CCK/oxy model may be expected to function over an exceptionally wide range of conditions. This strongly implies that the submodels effectively capture the

Table 17. Absolute Errors from an Optimization Using Only 12% O2 Data in an Oxy-Fuel Environment Black Thunder 12% O2 24% O2 36% O2

Mean Absolute Error (K)

Max Error (K)

Pittsburgh 8

11 58 92 Max Error (K)

12% O2 24% O2 36% O2

North Antelope

5 38 69 Mean Absolute Error (K)

12% O2 24% O2 36% O2

13 11 49

39 29 69

Mean Absolute Error (K)

Max Error (K)

Utah Skyline

9 17 31 Mean Absolute Error (K)

16 46 58 Max Error (K)

12% O2 24% O2 36% O2

35 31 49

53 60 89

from the data corresponds to the extrapolated predictions. It is necessary to decouple these two questions because even a very small shift in the time axis can cause a large shift in temperature for very early or very late particle burnout, which gives extremely misleading results. For example, the first data point of the Black Thunder coal in the 36% O2 extrapolation is offset from the data by approximately 2 ms, which results in an error between prediction and data of roughly 160 K, which far exceeds any of the other error values in the extrapolation. Furthermore, that two milliseconds of time disappear with a slight shift of diameter or gas temperature profile (well within the range of uncertainty), and the true particle temperatures span a wide range due to slight variations in maceral character, heat capacity, or particle shape. 4. A more accurate evaluation acknowledges that there is significant variation in the exact timing of initial particle heat-up and in the exact time of particle near-extinction. M

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Figure 5. Predictions in Conventional Conditions from 12% O2 Data Only.

necessary physics to make CCK/oxy a powerful predictive tool. However, it must be emphasized that the data here are a relatively small sample size of only four coals, and the current results would benefit from further validation with a wider range of data. Also, it is highly desirable to collect as much data at as many conditions as possible to minimize the type of bias observed in the Black Thunder and North Antelope results. 4.6. Extrapolation from 12% N2 Data. Because oxy-coal combustion has only become a popular research topic relatively recently, most literature data are not only obtained over a relatively narrow (and low) O2 range, but are also almost always in a conventional regime using N2 as the diluent. The CCK/oxy model would therefore be of most use if it could reasonably be calibrated from data collected at conventional conditions. Fortunately, the literature oxy-coal data used here were collected in parallel with data of the same coals at the same O2 concentration. The two experimental conditions differed only in that the second set of experiments used N2 as the diluent. In general, the method outlined in Section 4.5 applies to the optimizations that resulted in Figure 5 and Table 18, but differences and important similarities are highlighted below: The optimizations in this section were carried out using N2 data and conditions as inputs to the CCK/oxy model, but the plots and tables are extrapolations that took the kinetic parameters obtained from the 12% N2 data optimizations (Table 19) and applied them to all three O2 conditions in the oxy-fuel environment. 1. Table 20 shows the diameter values relevant to this set of experiments. 2. As in bullet point 2 in Section 4.5, four replicate optimizations were run for each set of data, and all had excellent agreement with each other except for one of the Utah Skyline replicates. This exception found a local minimum substantially farther from the minima found by the other three replicates (i.e., the optimization routine

Table 18. Absolute Errors from an Optimization Using Only 12% O2 Data in a Conventional Coal Environment Black Thunder

Mean Absolute Error (K)

Max Error (K)

Pittsburgh 8

15 50 84 Max Error (K)

12% O2 24% O2 36% O2

North Antelope

5 31 57 Mean Absolute Error (K)

12% O2 24% O2 36% O2

19 19 83

44 43 102

12% O2 24% O2 36% O2

Mean Absolute Error (K)

Max Error (K)

Utah Skyline

N/A N/A N/A Mean Absolute Error (K)

N/A N/A N/A Max Error (K)

12% O2 24% O2 36% O2

55 109 135

114 190 181

Table 19. Kinetic Parameters for the 12% O2 Condition in the Conventional Air-Fired Regime Parameter

Black Thunder

North Antelope

Pittsburgh 8

Utah Skyline

k3 (mol/cm3) k7 (mol/cm3) ER,3 (kJ/mol)

1.00 × 108 2.20 × 109 175

1.00 × 108 1.00 × 108 180

N/A N/A N/A

2.21 × 106 9.41 × 106 179

Table 20. Diameter Profiles for N2 Experiments

N

Coal Type

Black Thunder

North Antelope

Utah Skyline

O2 %

12

12

12

Height (cm)

Diameter (μm)

Diameter (μm)

Diameter (μm)

4.45 5.08 5.72 6.35 7.62 8.89 10.16 11.43 12.70

100 100 100 100 100

110

94 98 96 92 94 97

102 99 99

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3.

4.

5.

6.

7.

failed to find a reasonable optimum in one instance), and the results from the exception were discarded. The Pittsburgh coal data were unique in that the particle temperatures in N2 diluent were actually substantially lower than the particle temperatures in CO2 diluent. This one data set is inconsistent with theory, past experience, and all of the other 23 experimental data sets referenced here. The author of the paper containing the inconsistent Pittsburgh coal data theorized that this was because the gas temperature profile (which cannot be perfectly regulated) was colder in the N2 environment than in the CO2 environment.4 This is true, but even when accounting for the difference in profile temperature, the particles in the N2 environment are predicted to be roughly 50 K hotter than the CO2 environment, due to slower O2 diffusion and an endothermic gasification reaction in the CO2 environment. In the N2 environment data, the particles were roughly 150 K colder than in the CO2 environment for Pittsburgh coal. Adding in the prediction that they should be 50 K hotter, the total discrepancy is on the order of 200 K. Lacking a reasonable explanation for these anomalous data, the Pittsburgh coal data from the N2 experiment are not shown here. As before, the first point of the Black Thunder data and the last point of the North Antelope data are slightly offset in time, and not included in the averages for Table 18. The Black Thunder and North Antelope coals have the same bias toward underprediction of the particle temperature as before, and they are in general quite similar to the predictions from the extrapolations based on the 12% O2 in oxy-fuel conditions. This implies that there is no significant O2-char reaction mechanism change between oxy-coal and conventionally fired coal. The N2 Utah Skyline coal results did not extrapolate well to the oxy-fuel data, which could indicate either (1) that there is a mechanism change between the oxy-coal and conventional environments, (2) that the N2 Utah Skyline data are erroneous, or (3) that the exceptionally small N2 Utah Skyline data set (only 3 data collection heights) is insufficient to accurately capture the combustion kinetics. The last point (option 3) is thought to be the most likely explanation. As a whole, the extrapolation from conventional data at low O2 concentrations to the full range of oxy-fuel data has some promise but is far from conclusive. Further validation (additional data) is needed.

normal after sieving, swelling, and fragmentation, but the observed data are likely to be a truncated normal distribution. This is because the small diameter and/or rapidly oxidizable portion of the distribution quickly becomes undetectable due to the small radiative emission from these particles. The exact truncation point in the normal diameter distribution is unknown, and it is not consistent between burner heights, coal type, or O2 condition. To fully describe the true distribution of a data set, a correlation between particle detectability, temperature, and diameter would be devised. Accurate parameters for such a correlation would assume a char emissivity, require an assumed distribution for the raw coal, and incorporate knowledge of the optical limits of the detecting system. These assumptions, in conjunction with the coal swelling model, would predict the postdevolatilization diameter of a char particle, and the partially burned diameter at a given height, and appropriate correlation parameters would reconstruct the entire raw coal input diameter distribution. The second distribution of interest is the change in particle combustion behavior due to maceral character and ash content. For a given particle diameter, the combustion temperatures (from the literature data referenced in this work) vary by approximately ±150 K. If these values are simply used as error bars, the high accuracy of the CCK/oxy model is effectively useless. Instead, this variation should not be treated as error, but as actual variation in any given cohort of particles. The data shown here imply that a normal distribution with a mean of the CCK/oxy temperature prediction and a variance of approximately 75 K may be appropriate to capture the particle-toparticle variation. An appropriate CFD application to combine accuracy and computational efficiency is needed. One potential method would be to first determine the initial particle diameter distribution. Given an approximation of that distribution, CCK/oxy can be executed using n diameters that cover the distribution in sufficient detail. Bin values separated by 10 μm are likely adequate. For each bin, CCK/oxy should be executed with a gamut of gas temperature and composition profiles and with the output vectors recorded. Finally, the output vectors for a given bin size would be used to train a surrogate function that depends on gas composition, temperature, and the peak temperature in the burnout history of the particle. Such a function would execute very rapidly but potentially capture the majority of the information on the CCK/oxy model. Implementation into a CFD simulation would appropriately weight the available particle diameters and temperature variation within each diameter according to the two distributions described above.

6. CONCLUSIONS A comprehensive coal char conversion model (Carbon Conversion Kinetics) was extended to function at the extremes of oxy-coal combustion environments. These extensions included both numerical stability and submodel accuracy, including improved submodels for coal devolatilization, the mode of burning parameter, coal particle swelling, and the thermal annealing model. These improvements are thought to be valid in any char combustion regime, rather than being limited to oxy-coal combustion specifically. The specific submodel improvements given here were previously indicated to be the most sensitive submodels in a comprehensive, global sensitivity analysis.7 Model improvements were implemented, and the model was subsequently validated and explored by

5. APPLICATION TO CFD SIMULATIONS The literature data and the results of fitting the CCK/oxy model to data imply that a CFD simulation should take into account two distributions to accurately capture coal char particles. The first distribution is the diameter distribution of the raw coal particles that form the char. The diameter heavily impacts burnout predictions, and the mean, variance, and distribution form may propagate that impact on to the CFD simulation. In general, pulverized coal particle diameter distributions follow a Rosin−Rammler distribution, and it is this distribution that should be used in simulating industrial pulverized coal systems. For the experimental literature data used here, the full distribution is expected to be approximately O

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optimizing the model oxidation and gasification parameters to match a selection of the highly limited oxy-coal data from the literature. The validation revealed the following: 1. The CCK/oxy model matched the available data extremely well, with enormous improvement over past attempts using the CCK model.7,8 The CCK/oxy model was able to simultaneously fit all O2 conditions for a given coal with a single set of kinetic parameters. This was largely due to improvements in the devolatilization, swelling, and mode of burning models, as well as more exacting numerical solutions. The thermal annealing model is also exceptionally sensitive, but it is so tightly coupled to the kinetic preexponential factor that the submodel has minimal impact when optimizing the kinetic parameters of a single coal in a narrow range of heating rates and peak temperatures. Instead, the annealing model is vitally important to any attempt to create coal−general kinetic correlations or in exploring widely varying heating rate and peak temperature regimes with a given coal. 2. The CCK/oxy model, when optimized to the 12% O2 oxy-coal data only, made reasonable extrapolations to 24 and 36% O2 conditions. 3. The CCK/oxy model, when optimized to the 12% O2 conventional fired condition, made reasonable extrapolations to all levels of oxy-coal firing in two of three cases. These results are inconclusive but imply that data collected in conventional firing conditions may be useful in determining kinetic parameters relevant to oxy-coal scenarios. 4. In oxy-fuel conditions, several competing effects complicate the combustion regime. These effects are mainly due to high concentrations of gasification reactants (especially CO2), high temperatures that accompany enhanced O2 levels, and a balance between endothermic and exothermic reactions. The CCK/oxy model predictions are as anticipated: (1) that O2 combustion is by far the dominant reaction pathway, (2) that gasification becomes relatively less important at more intense oxygen conditions, and (3) that gasification becomes relatively more important at high temperature. The last two effects are in competition, and the second effect proved dominant here. Finally, as the present work was intended to support predictive boiler design via computational fluid dynamics simulation, a brief suggestion for CFD application was outlined. In this work, it was observed that both particle diameter distributions and particle reactivity distributions are vitally important to accurately model predictions. As CFD work ideally models the entirety of both distributions, accurate descriptions of both distributions must be estimated as closely as possible. This estimation is problematic when data are collected via radiant particle detection, because certain subsections of the activity and size distribution fall below the lower temperature and size limit of detectability.



Disclaimer. This publication was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. The authors declare no competing financial interest. Release Number. This document is authorized for public release (LA-UR-16-29489).



ACKNOWLEDGMENTS This material is based upon work supported by the Department of Energy, National Nuclear Security Administration, under Award Number DE-NA0002375. Funding for this work was also provided by the Department of Energy through the Carbon Capture Simulation Initiative.



AUTHOR INFORMATION

Corresponding Author

*E-mail: tom_fl[email protected], phone: 1-801-422-6236. ORCID

Thomas H. Fletcher: 0000-0002-9999-4492 P

NOMENCLATURE Parameter = Description Ac = The preexponential factor for computing the CO/CO2 ratio Ad = The preexponential factor in the thermal annealing reaction Ap = Particle area (m2) AR,1−AR,8 = The preexponential factor for 8 reactions and 2 reverse reactions. These come largely from correlations and can be adjusted for specific data c0 = The number of stable bridges cHR = An NMR structure based swelling parameter cj = The jth coefficient of the NMR correlations Cp = The per mass heat capacity of the char particle (J/kg/ K) dp,0 = The initial particle diameter, in microns EA = The activation energy in the annealing reaction Ec = The activation energy for computing the CO/CO2 ratio in cal/mol ER,1−ER,8 = The activation energy for 8 reactions and 2 reverse reactions. These come largely from correlations and can be adjusted for specific data f i = The fraction of active sites in bin i in the thermal annealing model HHR = Higher Heating Rate HR = The initial heating rate of the raw coal particle (K/s) ki = The rate constant of reaction i in the particle mp = The mass of the char particle (kg) Mδ = The average molecular weight of the side chains in a coal “monomer” Mcl = The average molecular weight of an aromatic cluster in a coal “monomer” N = The intrinsic order of R2 (formation of CO2 by combustion). This defaults to unity but can be adjusted to explore other kinetic regimes p0 = The fraction of intact bridges DOI: 10.1021/acs.energyfuels.6b03387 Energy Fuels XXXX, XXX, XXX−XXX

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(13) Shurtz, R. C.; Kolste, K. K.; Fletcher, T. H. Coal swelling model for high heating rate pyrolysis applications. Energy Fuels 2011, 25 (5), 2163−2173. (14) McBride, B.; Zehe, M.; Gordon, S. Nasa glenn coefficients for calculating thermodynamic properties of individual species; John H. Glenn Research Center: Cleveland, OH, 2002; p 291. (15) Rowley, R. L.; Wilding, W. V.; Oscarson, J. L.; Yang, Y.; Giles, N. F. Dippr® data compilation of pure chemical properties. In Design Institute for Physical Properties, AIChE; Design Institute for Physical Properties, AIChE: New York, NY, 2010. (16) Haugen, N. E. L.; Mitchell, R. E.; Tilghman, M. B. A comprehensive model for char particle conversion in environments containing O2 and CO2. Combust. Flame 2015, 162 (4), 1455−1463. (17) Haugen, N. E. L.; Tilghman, M. B.; Mitchell, R. E. The conversion mode of a porous carbon particle during oxidation and gasification. Combust. Flame 2014, 161 (2), 612−619. (18) Skokova, K. A. Selectivity in the carbon-oxygen reaction; Pennsylvania State University: 1997. (19) Sun, J. K.; Hurt, R. H. Mechanisms of extinction and nearextinction in pulverized solid fuel combustion. Proc. Combust. Inst. 2000, 28, 2205−2213. (20) Shurtz, R. Effects of pressure on the properties of coal char under gasification conditions at high initial heating rates. Chemical Engineering; Brigham Young University: 2011. (21) Niksa, S.; Liu, G. S.; Hurt, R. H. Coal conversion submodels for design applications at elevated pressures. Part i. Devolatilization and char oxidation. Prog. Energy Combust. Sci. 2003, 29, 425−477. (22) Liu, G. S.; Niksa, S. Coal conversion submodels for design applications at elevated pressures. Part ii. Char gasification. Prog. Energy Combust. Sci. 2004, 30, 679−717. (23) Hurt, R. H.; Calo, J. M. Semi-global intrinsic kinetics for char combustion modeling. Combust. Flame 2001, 125, 1138−1149. (24) McKay, M. D.; Beckman, R. J.; Conover, W. J. A comparison of three methonds for selecting values of input variabls in the analysis of output from a computer code. Technometrics 1979, 21 (2), 239−245. (25) Fletcher, T. H.; Kerstein, A. R.; Pugmire, R. J.; Solum, M. S.; Grant, D. M. Chemical percolation model for devolatilization 0.3. Direct use of C-13 NMR data to predict effects of coal type. Energy Fuels 1992, 6 (4), 414−431. (26) Grant, D. M.; Pugmire, R. J.; Fletcher, T. H.; Kerstein, A. R. Chemical model of coal devolatilization using percolation lattice statistics. Energy Fuels 1989, 3 (2), 175−86. (27) Genetti, D.; Fletcher, T. H.; Pugmire, R. J. Development and application of a correlation of C-13 NMR chemical structural analyses of coal based on elemental composition and volatile matter content. Energy Fuels 1999, 13 (1), 60−68. (28) Genetti, D. An advanced model of coal devolatilization based on chemical structure. Chemical Engineering; Brigham Young University: Provo, UT, 1999. (29) Hurt, R. H.; Mitchell, R. E. On the combustion kinetics of heterogeneous char particle populations; ACS Division of Fuel Chemistry: 1991; Vol. 202, pp 951−959. (30) Essenhigh, R. H. Influence of initial particle density on the reaction mode of porous carbon particles. Combust. Flame 1994, 99 (2), 269−279. (31) Smith, I. W. The combustion rates of coal chars: A review. Symp. (Int.) Combust., [Proc.] 1982, 19, 1045−1065. (32) Mitchell, R. E.; Hurt, R. H.; Baxter, L. L.; Hardesty, D. R. Compilation of sandia coal char combustion data and kinetic analyses. Milestone report; Sandia National Laboratories: Livermore, CA, 1992; p 615. (33) Oh, M. S.; Peters, W. A.; Howard, J. B. An experimental and modeling study of softening coal pyrolysis. AIChE J. 1989, 35 (5), 775−792. (34) Yang, H.; Li, S.; Fletcher, T. H.; Dong, M. Simulation of the swelling of high-volatile bituminous coal during pyrolysis. Energy Fuels 2014, 28 (11), 7216−7226. (35) Yang, H.; Li, S.; Fletcher, T. H.; Dong, M. Simulation of the swelling of high-volatile bituminous coal during pyrolysis. Part 2:

Pi = The partial pressure of reactive gas i at the surface of the particle rp,i = The rate of reaction i in the particle smin = A proximate analysis based swelling parameter svar = An NMR structure based swelling parameter Tg = The gas temperature (K) Tp = The particle temperature (K) Ts = The temperature of the surroundings for radiative heat transfer (K) Xc = The percentage carbon from the ultimate analysis α = The mode of burning parameter ΔHrxn,i = The enthalpy of reaction i in the particle εp = Particle emissivity η = The effectiveness factor Φi = The Thiele modulus ψ = A random pore model parameter. This value has some uncertainty, and defaults to 4.6 σ = Stefan−Boltzmann constant or a parameter in the log− normal distribution σ+1 = The coordination number τ/f = A random pore model parameter. This value has some uncertainty, and defaults to 12 Ω = The swelling coefficient (dp/dp,0)



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R

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